If you're looking for anime similar to The Morose Mononokean, you might like these titles.
When Miyako Arata joins the Shinjuku Ward Office, he thinks he's gotten a normal civil servant job. But it turns out he's joined the Night Community Exchange Department, one of which operates secretly in each of Tokyo's twenty-three ward offices. Their job is to resolve occult issues concerning non-human beings. Accompanied by his senpai and department head Sakaki Kyoichi and the occult obsessed Himetsuka Seo, they work night after night, facing off with beings whose existence defies the laws of our world.
Both of these series feature people who are in the professional business of dealing with youkai and other supernatural beings.
In feudal Japan, evil spirits known as mononoke plague both households and the countryside, leaving a trail of fear in their wake. One mysterious person has the power to slay the mononoke where they stand; he is known only as the Medicine Seller, and he vanquishes the mononoke using the power of his Exorcism Sword. However, in order to draw his sword he must first understand the Form, Truth and Reason of the mononoke. Armed with a sharp wit and keen intellect, the Medicine Seller wanders from place to place, striking down the mononoke in his wake.